An improved correction of radial-velocity systematic for the SOPHIE spectrograph

High precision spectrographs might exhibit temporal variations of their reference velocity or nightly zero point (NZP). One way to monitor the NZP is to measure bright stars, which are assumed to have an intrinsic radial velocity variation much smaller than the instrument's precision. While this method is effective in most cases, it does not fully propagate the uncertainty arising from NZP variations. We present a new method to correct for NZP variations in radial-velocity time series. This method uses Gaussian Processes based on ancillary information to model these systematic effects. It enables us to propagate the uncertainties of this correction into the overall error budget. Another advantage of this approach is that it relies on ancillary data collected simultaneously with the spectra rather than solely on dedicated observations of constant stars. We applied this method to the SOPHIE spectrograph at the Haute-Provence Observatory using a few instrument's housekeeping data, such as the internal pressure and temperature variations. Our results demonstrate that this method effectively models the red noise of constant stars, even with a limited amount of housekeeping data, while preserving the signals of exoplanets. Using both simulations with mock planets and real data, we found that this method improves the false-alarm probability of detections by several orders of magnitude. By simulating numerous planetary signals, we were able to detect up to 10 percent more planets with small amplitude radial velocity signals. We used this new correction to reanalysed the planetary system around HD158259 and improved the detection of the outermost planets. We also suggest decreasing the observing cadence of the constant stars to optimise telescope time for scientific targets.